Weighted anisotropic product Hardy spaces and boundedness of sublinear operators

Abstract

Abstract Let A 1 and A 2 be expansive dilations, respectively, on ℝ n and ℝ m . Let A ≡ ( A 1 , A 2 ) and 𝒜 p ( A ) be the class of product Muckenhoupt weights on ℝ n × ℝ m for p ∈ (1, ∞]. When p ∈ (1, ∞) and w ∈ 𝒜 p ( A ), the authors characterize the weighted Lebesgue space L p w (ℝ n × ℝ m ) via the anisotropic Lusin‐area function associated with A . When p ∈ (0, 1], w ∈ 𝒜 ∞ ( A ), the authors introduce the weighted anisotropic product Hardy space H p w (ℝ n × ℝ m ; A ) via the anisotropic Lusin‐area function and establish its atomic decomposition. Moreover, the authors prove that finite atomic norm on a dense subspace of H p w (ℝ n ×ℝ m ; A ) is equivalent with the standard infinite atomic decomposition norm. As an application, the authors prove that if T is a sublinear operator and maps all atoms into uniformly bounded elements of a quasi‐Banach space ℬ︁, then T uniquely extends to a bounded sublinear operator from H p w (ℝ n × ℝ m ; A ) to ℬ︁. The results of this paper improve the existing results for weighted product Hardy spaces and are new even in the unweighted anisotropic setting (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)